Write a single congruence?

Question

Write a single congruence that is equivalent to the pair of congruences:

$x\equiv 1(\mod4)$ and $x\equiv 2 (\mod 3)$.

I am new to Number Theory/ Modular Arithmetic. Just started reading the theory from a book yesterday. Earlier I had read about it a little on the internet. I know the basic definitions and some properties. I do not understand the question properly. What does writing a single congruence mean? Does it mean I have to find $x$? How do I solve this question. Do I need to use any theorem like the Chinese Remainder Theorem or something else? I am stuck. Please help.

Answer 1

Hint:

Start from a Bézout's relation between the moduli, say $\; 4-3=1$, and consider the analog of the linear interpolation polynomial: for any $a, b \in\mathbf Z$, $$a\cdot 4-b\cdot3\equiv \begin{cases}a \mod 3, \\[1ex] b\mkern 1.5mu \mod 4. \end{cases}$$

Answer 2

you can literally turn this into $3z+2=4y+1;3z+1=4y$; take this mod 3 and get 1=y; so plug $y=3a+1$ and get back $3z+1=12a+4$ and then add 1 to both sides restoring the original getting $3z+2=12a+5$ so you get $x\equiv 5 \bmod 12$. This is basically CRT done via linear polynomials.